1. **State the problem:** We are given a right rectangular prism with three face areas: front face area = 39 cm², top face area = 65 cm², and side face area = 15 cm². We need to find the whole-number dimensions of the prism and then calculate its volume. 2. **Define variables:** Let the dimensions be $l$ (length), $w$ (width), and $h$ (height). 3. **Write equations from the face areas:** - Front face area = $l \times h = 39$ - Top face area = $l \times w = 65$ - Side face area = $w \times h = 15$ 4. **Use the equations to find dimensions:** From $l \times h = 39$ and $l \times w = 65$, express $l$: $$ l = \frac{39}{h} = \frac{65}{w} $$ Equate the two expressions for $l$: $$ \frac{39}{h} = \frac{65}{w} $$ Cross-multiply: $$ 39w = 65h $$ Simplify by dividing both sides by 13: $$ \cancel{13} \times 3 w = \cancel{13} \times 5 h \Rightarrow 3w = 5h $$ 5. **Express $w$ in terms of $h$:** $$ w = \frac{5h}{3} $$ 6. **Use the side face area $w \times h = 15$:** Substitute $w$: $$ \left(\frac{5h}{3}\right) \times h = 15 $$ $$ \frac{5h^2}{3} = 15 $$ Multiply both sides by 3: $$ 5h^2 = 45 $$ Divide both sides by 5: $$ h^2 = 9 $$ Take the positive root (since dimensions are positive): $$ h = 3 $$ 7. **Find $w$ using $w = \frac{5h}{3}$:** $$ w = \frac{5 \times 3}{3} = 5 $$ 8. **Find $l$ using $l \times h = 39$:** $$ l = \frac{39}{h} = \frac{39}{3} = 13 $$ 9. **Check all areas:** - Front face: $13 \times 3 = 39$ (correct) - Top face: $13 \times 5 = 65$ (correct) - Side face: $5 \times 3 = 15$ (correct) 10. **Calculate volume:** $$ V = l \times w \times h = 13 \times 5 \times 3 = 195 $$ **Final answers:** - Dimensions: $13$ cm (length), $5$ cm (width), $3$ cm (height) - Volume: $195$ cm³